We present a method to generate many automorphisms of a supersingular K3 surface in odd characteristic. As an application, we show that, if p is an odd prime less than or equal to 7919, then every supersingular K3 surface in characteristic p has an automorphism whose characteristic polynomial on the Néron-Severi lattice is a Salem polynomial of degree 22. For a supersingular K3 surface with Artin invariant 10, the same holds for odd primes less than or equal to 17389.
In this note we exhibit explicit automorphisms of maximal Salem degree 22 on the supersingular K3 surface of Artin invariant one for all primes p congruent 3 mod 4 in a systematic way. Automorphisms of Salem degree 22 do not lift to any characteristic zero model.
We give a short proof that every supersingular K3 surface (except possibly in characteristic $2$ with Artin invariant $σ=10$) has an automorphism of Salem degree 22. In particular an infinite subgroup of the automorphism group does not lift to characteristic zero. The proof relies on the case $σ=1$ and the cone conjecture for K3 surfaces.
Using elliptic structures, we show that any supersingular K3 surface of Artin invariant $1$ in characteristic $p \not= 5$, $7$, $13$ has an automorphism the entropy of which is the natural logarithm of a Salem number of degree $22$.
In this article we give a strategy to decide whether the logarithm of a given Salem number is realized as entropy of an automorphism of a supersingular K3 surface in positive characteristic. As test case it is proved that $\log λ_d$, where $λ_d$ is the minimal Salem number of degree $d$, is realized in characteristic $5$ if and only if $d\leq 22$ is even and $d\neq 18$. In the complex projective setting we settle the case of entropy $\log λ_{12}$ left open by McMullen, by giving the construction. A necessary and sufficient test is developed to decide whether a given isometry of a hyperbolic la…
We study the maximal Salem degree of automorphisms of K3 surfaces via elliptic fibrations. By generalizing \cite{EOY14}, we establish a characterization of such maximum in terms of elliptic fibrations with infinite automorphism groups. As an application, we show that any supersingular K3 surface in odd characteristic has an automorphism the entropy of which is the natural logarithm of a Salem number of degree $22$.
If an automorphism of a projective K3 surface with Picard number 2 is of infinite order, then the automorphism corresponds to a solution of Pell equation. In this paper, by solving this equation, we determine all Salem polynomials of symplectic and anti-symplectic automorphisms of projective K3 surfaces with Picard number 2.
We show that the Mathieu groups $M_{22}$ and $M_{11}$ can act on the supersingular $K3$ surface with Artin invariant 1 in characteristic 11 as symplectic automorphisms. More generally we show that all maximal subgroups of the Mathieu group $M_{23}$ with three orbits on 24 letters act on a supersingular $K3$ surface with Artin invariant 1 in a suitable characteristic.
It is known that the dynamical degree, or equivalently, the topological entropy of an automorphism g of an algebraic surface S is lower semi-continuous when (S,g) varies in a algebraic family. In this paper we make a series of experiments confirming this behavior with the aim to realize small Salem numbers as the dynamical degrees of automorphisms of Enriques surfaces.
In characteristic $p=0$ or $p>5$, we show that a K3 surface with an order 60 automorphism is unique up to isomorphism. As a consequence, we characterize the supersingular K3 surface with Artin invariant 1 in characteristic $p=11$ (mod 12) by a cyclic symmetry of order 60.
The automorphism group of a supersingular K3 surface with Artin invariant 1 in characteristic 3
Graduate School of Mathematics, Nagoya University（名古屋大学大学院数学研究科）；Department of Mathematics, Graduate School of Science, Hiroshima University（广岛大学理学研究科数学系）
This article extends Bayer-Fluckiger's theorem on characteristic polynomials of isometries on an even unimodular lattice to the case where the isometries have determinant $-1$. As an application, we show that the logarithm of every Salem number of degree $20$ is realized as the topological entropy of an automorphism of a nonprojective K3 surface.
We realize the logarithm of the third smallest known Salem number as the topological entropy of a K3 surface automorphism with a Siegel disk and a pointwisely fixed curve at the same time. We also show the logarithm of the Lehmer number, the smallest known Salem number, is not realizable as the topological entropy of any Enriques surface automorphism. These results are entirely inspired by McMullen's works and Mathematica programs.
We classify two-dimensional complex tori admitting automorphisms with positive entropy in terms of the entropies they exhibit. For each possible positive value of entropy, we describe the set of two-dimensional complex tori admitting automorphisms with that entropy.
We show that, in any dimension greater than one, there are an abelian variety, a smooth rational variety and a Calabi-Yau manifold, with primitive birational automorphisms of first dynamical degree $>1$. We also show that there are smooth complex projective Calabi-Yau manifolds and smooth rational manifolds, of any even dimension, with primitive biregular automorphisms of positive topological entropy.
For two-dimensional complex tori, we characterize the set of all values of positive entropy that arise from automorphisms. For K3 surfaces, we give suffcient conditions for a positive value to be the entropy of some automorphism.
Let X and Y be supersingular K3 surfaces defined over an algebraically closed field. Suppose that the sum of their Artin invariants is 11. Then there exists a certain duality between their Néron-Severi lattices. We investigate geometric consequences of this duality. As an application, we classify genus one fibrations on supersingular K3 surfaces with Artin invariant 10 in characteristic 2 and 3, and give a set of generators of the automorphism group of the nef cone of these supersingular K3 surfaces. The difference between the automorphism group of a supersingular K3 surface X and the automorp…
For any odd characteristic p=2 mod 3, we exhibit an explicit automorphism on the supersingular K3 surface of Artin invariant one which does not lift to any characteristic zero model. Our construction builds on elliptic fibrations to produce a closed formula for the automorphism's characteristic polynomial on second cohomology, which turns out to be an irreducible Salem polynomial of degree 22 with coefficients varying with p.
We construct explicitly moduli curves of polarized supersingular K3 surfaces in characteristic 2 with Artin invariant 2. As an application, we detect a "jump" phenomenon in a family of automorphism groups of supersingular K3 surfaces with a constant Néron-Severi lattice.
Consider an arbitrary automorphism of an Enriques surface with its lift to the covering K3 surface. We prove a bound of the order of the lift acting on the anti-invariant cohomology sublattice of the Enriques involution. We use it to obtain some mod 2 constraint on the original automorphism. As an application, we give a necessary condition for Salem numbers to be dynamical degrees on Enriques surfaces and obtain a new lower bound on the minimal value. In the Appendix, we give a complete list of Salem numbers that potentially may be the minimal dynamical degree on Enriques surfaces and for whic…
We show that a characteristic $0$ model $X_R\to \Spec R$, with Picard number $1$ over a geometric generic point, of a K3 surface in characteristic $p\ge 3$, essentially kills all automorphisms (Theorem 5.1). We show that there is an explicitely constructed automorphism on a supersingular K3 surface in characteristic $3$, which has positive entropy, the logarithm of a Salem number of degree $22$ (Theorem 6.4). In particular it does not lift to characteristic $0$. In addition, we show that in any large characteristic, there is an automorphism of a supersingular K3 which has positive entropy and …
We present three interesting projective models of the supersingular K3 surface X in characteristic 5 with Artin invariant 1. For each projective model, we determine smooth rational curves on X with the minimal degree and the projective automorphism group. Moreover, by using the superspecial abelian surface, we construct six sets of 16 disjoint smooth rational curves on X, and show that they form a beautiful configuration.
Let $X$ be a compact Kähler manifold of dimension $k\leq 4$ and $f:X\rightarrow X$ a pseudo-automorphism. If the first dynamical degree $λ_1(f)$ is a Salem number, we show that either $λ_1(f)=λ_{k-1}(f)$ or $λ_1(f)^2=λ_{k-2}(f)$. In particular, if $\mbox{dim}(X)=3$ then $λ_1(f)=λ_2(f)$. We use this to show that if $X$ is a complex 3-torus and $f$ is an automorphism of $X$ with $λ_1(f)>1$, then $f$ has a non-trivial equivariant holomorphic fibration if and only if $λ_1(f)$ is a Salem number. If $X$ is a complex 3-torus having an automorphism $f$ with $λ_1(f)=λ_2(f)>1$ but is not a Salem number,…
Let p be a prime integer, and q a power of p. The Ballico-Hefez curve is a non-reflexive nodal rational plane curve of degree q+1 in characteristic p. We investigate its automorphism group and defining equation. We also prove that the surface obtained as the cyclic cover of the projective plane branched along the Ballico-Hefez curve is unirational, and hence is supersingular. As an application, we obtain a new projective model of the supersingular K3 surface with Artin invariant 1 in characteristic 3 and 5.
For any given Salem number, we construct an automorphism on a simple abelian variety whose first dynamical degree is the square of the Salem number. Our construction works for both simple abelian varieties with totally indefinite quaternion multiplication and for simple abelian varieties of the second kind. We then give a complete classification of the dynamical degree sequences for abelian varieties of dimension at most four and obtain an ergodic result for sequences of pullbacks of forms.
Using the theory of polarised flag type quotients, we determine mass formulae for all principally polarised supersingular abelian threefolds defined over an algebraically closed field $k$ of characteristic $p$. We combine these results with computations of the automorphism groups to study Oort's conjecture; we prove that every generic three-dimensional principally polarised supersingular abelian variety over $k$ of characteristic $\neq 2$ has automorphism group $\{ \pm 1 \}$.
We construct a K3 surface over an algebraically closed field of characteristic 2 which contains two sets of 21 disjoint smooth rational curves such that each curve from one set intersects exactly 5 curves from the other set. This configuration is isomorphic to the configuration of points and lines on the projective plane over the finite field of 4 elements. The surface admits a finite automorphism group which contains the group of automorphisms of the plane which acts on the configuration of each set of 21 smooth rational curves, and the additional element of order 2 which interchanges the two…
This paper concerns K3 surfaces with automorphisms of order 11 in arbitrary characteristic. Specifically we study the wild case and prove that a general such surface in characteristic 11 has Picard number 2. We also construct K3 surfaces with an automorphism of order 11 in every characteristic, and supersingular K3 surfaces whenever possible.
Let X be a supersingular K3 surface in characteristic 5 with Artin invariant 1. Then X has a polarization that realizes X as the Fermat sextic double plane. We present a list of polarizations of X with degree 2 whose intersection number with this Fermat sextic polarization is less than or equal to 5, and give the defining equations of the corresponding projective models. We also present a method to describe birational morphisms between these projective models explicitly. As a by-product, a non-projective automorphism of the Fermat sextic double plane is obtained.
Complex Enriques surfaces with a finite group of automorphisms are classified into seven types. In this paper, we determine which types of such Enriques surfaces exist in characteristic 2. In particular we give a one dimensional family of classical and supersingular Enriques surfaces with the automorphism group isomorphic to the symmetric group of degree five.
We classify supersingular and classical Enriques surfaces with finite automorphism group in characteristic 2 into 8 types according to their dual graphs of all $(-2)$-curves (nonsigular rational curves). We give examples of these Enriques surfaces together with their canonical coverings. It follows that the classification of all Enriques surfaces with finite automorphism group in any characteristics has been finished.
An automorphism of an algebraic surface $S$ is called cohomologically (numerically) trivial if it acts identically on the second $l$-adic cohomology group (this group modulo torsion subgroup). Extending the results of S. Mukai and Y. Namikawa to arbitrary characteristic $p > 0$, we prove that the group of cohomologically trivial automorphisms $\rm{Aut}_{\rm{ct}}(S)$ of an Enriques surface $S$ is of order $\leq 2$ if $S$ is not supersingular. If $p = 2$ and $S$ is supersingular, we show that $\rm{Aut}_{\rm{ct}}(S)$ is a cyclic group of odd order $n\in \{1,2,3,5,7,11\}$ or the quaternion group $…
We characterize Salem numbers which have some power arising as dynamical degree of an automorphism on a complex (projective) 2-Torus, K3 or Enriques surface.
Let X be a complex algebraic K3 surface or a supersingular K3 surface in odd characteristic. We present an algorithm by which, under certain assumptions on X, we can calculate a finite set of generators of the image of the natural homomorphism from the automorphism group of X to the orthogonal group of the Néron-Severi lattice of X. We then apply this algorithm to certain complex K3 surfaces, among which are singular K3 surfaces whose transcendental lattices are of small discriminants.
In this paper, we prove that, over an algebraically closed field of odd characteristic, a weakly tame automorphism of a K3 surface of finite height can be lifted over the ring of Witt vectors of the base field. Also we prove that a non-symplectic tame automorphism of a supersingular K3 surface or a symplectic tame automorphism of a supersingular K3 surface of Artin-invariant at least 2 can be lifted over the ring of Witt vectors. Using these results, we prove, for a weakly tame K3 surface of finite height, there is a lifting over the ring of Witt vectors to which whole the automorphism group o…
We introduce and study a new way to catagorize supersingular abelian varieties defined over a finite field by classifying them as fully maximal, mixed, or fully minimal. The type of $A$ depends on the normalized Weil numbers of $A$ and its twists. We analyze these types for supersingular abelian varieties and curves under conditions on the automorphism group. In particular, we present a complete analysis of these properties for supersingular elliptic curves and supersingular abelian surfaces in arbitrary characteristic, and for a one-dimensional family of supersingular curves of genus $3$ in c…
The aim of this paper is to give necessary and sufficient conditions for an integral polynomial to be the characteristic polynomial of a semi-simple isometry of some even unimodular lattice of given signature. This result has applications applications to automorphisms of K3 surfaces; in particular, we show that every Salem number of degree 4, 6, 8,1 2, 14 or 16 is the dynamical degree of an automorphism of a non-projective K3 surface.
We construct Zariski K3 surfaces of Artin invariant 1, 2 and 3 in many characteristics. In particular, we prove that any supersingular Kummer surface is Zariski if the characteristic is not congruent to 1 modulo 12. Our methods combine different approaches such as quotients by the group scheme $α_p$, Kummer surfaces, and automorphisms of elliptic and hyperelliptic curves.
If a K3 surface admits an automorphism with a Siegel disk, then its Picard number is an even integer between $0$ and $18$. Conversely, using the method of hypergeometric groups, we are able to construct K3 surface automorphisms with Siegel disks that realize all possible Picard numbers. The constructions involve extensive computer searches for appropriate Salem numbers and computations of algebraic numbers arising from holomorphic Lefschetz-type formulas and related Grothendieck residues.
We study K3 surfaces with 9 cusps, i.e. 9 disjoint $A_2$ configurations of smooth rational curves, over algebraically closed fields of characteristic $p\neq 3$. Much like in the complex situation studied by Barth, we prove that each such surface admits a triple covering by an abelian surface. Conversely, we determine which abelian surfaces with order three automorphisms give rise to K3 surfaces. We also investigate how K3 surfaces with 9 cusps hit the supersingular locus.
We give several results concerning the connected component ${\rm Aut}_X^0$ of the automorphism scheme of a proper variety $X$ over a field, such as its behaviour with respect to birational modifications, normalization, restrictions to closed subschemes and deformations. Then, we apply our results to study the automorphism scheme of not necessarily Jacobian elliptic surfaces $f: X \to C$ over algebraically closed fields, generalizing work of Rudakov and Shafarevich, while giving counterexamples to some of their statements. We bound the dimension $h^0(X,T_X)$ of the space of global vector fields…
The elliptic modular surface of level 4 is a complex K3 surface with Picard number 20. This surface has a model over a number field such that its reduction modulo 3 yields a surface isomorphic to the Fermat quartic surface in characteristic 3, which is supersingular. The specialization induces an embedding of the Néron-Severi lattices. Using this embedding, we determine the automorphism group of this K3 surface over a discrete valuation ring of mixed characteristic whose residue field is of characteristic 3. The elliptic modular surface of level 4 has a fixed-point free involution that gives r…
We show that Mukai's classification of finite groups which may act symplectically on a complex K3 surface extends to positive characteristic $p$ under the assumptions that (i) the order of the group is coprime to $p$ and (ii) either the surface or its quotient is not birationally isomorphic to a supersingular K3 surface with Artin invariant 1. In the case without the assumption (ii) we classify all possible new groups which may appear. We prove that the assumption on the order of the group is always satisfied if $p > 11$ and if $p=2,3,5,11$ we give examples of K3 surfaces with finite symplecti…
For a K3 surface over an algebraically closed field of odd characteristic, the representation of the automorphism group on the global two forms is finite. If the K3 surface is supersingular, it is isomorphic to the representation on the discriminant group of the Neron-Severi group. If the K3 surface is of finite height, the representation on the (etale or crystalline) transcendental cycles is also finite and there is a canonical projection from the representation on the transcendental cycles to the representation on the two forms. We prove that, if the base field is an algebraic closure of a f…
